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The number of Latin squares of order 11


Authors: Alexander Hulpke, Petteri Kaski and Patric R. J. Östergård
Journal: Math. Comp. 80 (2011), 1197-1219
MSC (2010): Primary 05B15, 05A15, 05C30, 05C70
DOI: https://doi.org/10.1090/S0025-5718-2010-02420-2
Published electronically: September 13, 2010
MathSciNet review: 2772119
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Abstract: Constructive and nonconstructive techniques are employed to enumerate Latin squares and related objects. It is established that there are (i) $ 2036029552582883134196099$ main classes of Latin squares of order $ 11$$ ;$ (ii) $ 6108088657705958932053657$ isomorphism classes of one-factorizations of $ K_{11,11}$$ ;$ (iii) $ 12216177315369229261482540$ isotopy classes of Latin squares of order $ 11$$ ;$ (iv) $ 1478157455158044452849321016$ isomorphism classes of loops of order $ 11$$ ;$ and (v) $ 19464657391668924966791023043937578299025$ isomorphism classes of quasigroups of order $ 11$. The enumeration is constructive for the $ 1151666641$ main classes with an autoparatopy group of order at least $ 3$.


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Additional Information

Alexander Hulpke
Affiliation: Department of Mathematics, Colorado State University, 1874 Campus Delivery, Fort Collins, Colorado 80523-1874
Email: hulpke@math.colostate.edu

Petteri Kaski
Affiliation: Helsinki Institute for Information Technology HIIT, University of Helsinki, Department of Computer Science, P.O. Box 68, 00014 University of Helsinki, Finland
Email: petteri.kaski@cs.helsinki.fi

Patric R. J. Östergård
Affiliation: Department of Communications and Networking, Aalto University, P.O. Box 13000, 00076 Aalto, Finland
Email: patric.ostergard@tkk.fi

DOI: https://doi.org/10.1090/S0025-5718-2010-02420-2
Keywords: Enumeration, Latin square, main class
Received by editor(s): September 18, 2009
Received by editor(s) in revised form: February 4, 2010
Published electronically: September 13, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.